Spin waves and spin diffusion in Fermi liquids: Bounds on effective diffusion coefficients

We investigate the accuracy of the usual relaxation-time approximations, involving the spin-diffusion lifetime ${\tau }_D$, which are generally made in analyses of spin waves and the Leggett-Rice effect in Fermi liquids. By employing the variational methods of Ah-Sam, Ho/jgaard-Jensen, and Smith and of Egilsson and Pethick, we are able to determine upper and lower bounds on the effective diffusion coefficient resulting from spin-wave phenomena which are accurate in the whole Fermi-liquid regime (T≪$T_F$). Our results indicate that the usual approximations break down for T≲7 mK in ${}_{}{}^3{}_{}{}^{}\mathrm{He}$, but are accurate to within ∼2% in 5% ${}_{}{}^3{}_{}{}^{}\mathrm{-}_{}^4{}_{}{}^{}$He mixtures. Our results are consistent with values of $F_1^a$≊-0.6 at 0 bar, ≊-0.4 at 27 bars in ${}_{}{}^3{}_{}{}^{}\mathrm{He}$, but these have considerable uncertainties.